{"paper":{"title":"Reciprocals of Subsum Polynomials","license":"http://creativecommons.org/licenses/by/4.0/","headline":"The sum of reciprocals of subsum polynomials over all partitions of n has arithmetic properties and connections to combinatorial objects.","cross_cats":["math.CO"],"primary_cat":"math.NT","authors_text":"Brooke Feigon, Cristina Ballantine, George Beck, Kathrin Maurischat","submitted_at":"2026-05-11T13:04:47Z","abstract_excerpt":"We introduce the subsum polynomial of a partition $\\lambda=(\\lambda_1, \\lambda_2, \\ldots, \\lambda_k)$ defined by $\\mathrm{sp}(\\lambda, x)=\\prod_{i=1}^k(1+x^{\\lambda_i})$. We study the sum of reciprocals of $\\mathrm{sp}(\\lambda, x)$ over all partitions of $n$. We prove arithmetic properties of related polynomials and offer connections to other combinatorial objects."},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"We introduce the subsum polynomial of a partition λ=(λ1, λ2, …, λk) defined by sp(λ, x)=∏i=1k(1+x^λi). We study the sum of reciprocals of sp(λ, x) over all partitions of n. We prove arithmetic properties of related polynomials and offer connections to other combinatorial objects.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"That the sum of reciprocals of the subsum polynomials over partitions of n admits provable arithmetic properties and non-trivial connections to other combinatorial objects.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"Introduces the subsum polynomial sp(λ, x) = product (1 + x^λi) for partitions λ and studies the sum of reciprocals over all partitions of n, proving arithmetic properties and combinatorial connections.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"The sum of reciprocals of subsum polynomials over all partitions of n has arithmetic properties and connections to combinatorial objects.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"4bcaaf7ced1c53bce002fcc64a7ca49b71bee8d2b5600ef5d78e58d06171a2b5"},"source":{"id":"2605.10512","kind":"arxiv","version":2},"verdict":{"id":"d12105f0-7178-4a4f-a91e-94521a047898","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-12T05:24:57.070068Z","strongest_claim":"We introduce the subsum polynomial of a partition λ=(λ1, λ2, …, λk) defined by sp(λ, x)=∏i=1k(1+x^λi). We study the sum of reciprocals of sp(λ, x) over all partitions of n. We prove arithmetic properties of related polynomials and offer connections to other combinatorial objects.","one_line_summary":"Introduces the subsum polynomial sp(λ, x) = product (1 + x^λi) for partitions λ and studies the sum of reciprocals over all partitions of n, proving arithmetic properties and combinatorial connections.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"That the sum of reciprocals of the subsum polynomials over partitions of n admits provable arithmetic properties and non-trivial connections to other combinatorial objects.","pith_extraction_headline":"The sum of reciprocals of subsum polynomials over all partitions of n has arithmetic properties and connections to combinatorial objects."},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2605.10512/integrity.json","findings":[],"available":true,"detectors_run":[{"name":"claim_evidence","ran_at":"2026-05-20T05:42:01.027164Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"ai_meta_artifact","ran_at":"2026-05-19T14:42:25.429011Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"doi_title_agreement","ran_at":"2026-05-19T11:01:17.928691Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"doi_compliance","ran_at":"2026-05-19T09:13:29.089366Z","status":"completed","version":"1.0.0","findings_count":0}],"snapshot_sha256":"aff50329a3564ea35965d80973da5b8cb6b4c69cb7799871699797edbd96fc1f"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":2,"snapshot_sha256":"2a21ae4ebd221883cc0810b30908031f74202a131518660ecfaaa0db83a7af9a"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}